Two circles touch each other at point C. Prove that the common tangent to the circles at C, bisects the common tangent P & Q.

We know that tangents drawn from an external point to a circle are equal in length.

So, PA = PC  and  PB = PC

⇒PA = PB  so, P is the mid point of AB

Similarly, DQ = QC and QE = QC

⇒DQ = QE so, Q is the mid point of DE.

Thus, the common tangent at point C bisects the common tangents at P and Q.